CURVES  WITH  A  DIRECTRIX 


DISSERTATION 

Submitted  to  the  Board  of  University  Studies  of  the  Johns  Hopkins  University 

in  conformity  with  the  requirements  for  the  deg;ree  of 

Doctor  of  Philosophy 


BY 
CLYDE  SHEPHERD  ATCHISON 
J907 


BALTIMORE,  MD.,  V.  S.  A. 
1908 


To  Professor  Morley,  without  whose  helpful  suggestions  and  constant 
inspiration  this  paper  would  have  been  impossible,  and  to  Dr.  Cohen,  Dr. 
Hulbert,  and  Dr,  Coble  for  their  encouragement  in  his  university  course,  the 
author  desires  thus  to  express  his  thanks. 


186885 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/curveswithdirectOOatchrich 


/  OF  THE 

I    UNIVERSITY 


Curves  ivith  a  Directrix. 

Br  Clyde  S.  Atchison. 


Introductory. 

It  is  a  well-known  fact  of  elementary  geometry  that  the  orthocentre  of  any 
three  lines  of  a  parabola,  which  is  of  course  uniquely  determined  by  four  lines, 
lies  on  a  line,  called  the  directrix  of  the  parabola.  In  his  Ortliocentric  Properties 
of  the  Plane  n-Line,  Professor  Morlet*  has  extended  the  meaning  of  the  term 
orthocentre,  so  that  with  every  odd  number  of  lines  there  is  associated  an  ortho- 
centre,  and  with  every  even  number  of  lines  a  directrix.  It  is  my  purpose  to 
discuss  the  class  of  curves,  of  which  the  orthocentre  of  a  certain  number  of  lines 
shall  always  lie  on  a  line,  the  directrix  of  the  curve  in  question.  As  the  curves 
to  be  considered  are  entirely  rational,  throughout  the  discussion,  I  shall  employ 
almost  exclusively  the  method  of  vector  analysis,  worked  out  by  Professor 
MoRLEY,  which  is  especially  convenient  for  the  purpose.  The  line  equation  of 
a  curve  is  here  uniformly  expressed  by  means  of  conjugate  coordinatesf,  while 
the  point  equation  is  expressed  by  means  of  what  is  commonly  called  the  map 
equation ;  that  is,  a  point  of  the  curve  is  expressed  as  a  rational  algebraic  function 
of  a  parameter  t,  which  is  limited  to  the  unit  circle,  and  is,  therefore,  always 
equal  in  absolute  value  to  unity.  Throughout  y  and  h  will  be  considered  as 
conjugates  of  x  and  a  respectively;  that  is,  if  X  and  Y  are  rectangular  coordi- 
nates of  a  point,  then 

x=zX  -\-iY, 

y  =  X-iY, 
and  similarly  for  a  and  h.  { 

*  Transactions  of  the  American  Mathematical  Society,  Vol.  IV  (1903),  pp.  1-13. 

f  F.  Fbanklin:  Some  Applications  of  Circular  Coordinates,  ^mcWcan  Journal  of  Mathematics,  Vol.  XII 
(1890),  p.  161;  F.  Moklby  :  On  the  Metric  Geometry  of  the  Plane  n-Line,  Transactions  of  the  American  Society, 
Vol.  I  (1900),  p.  97. 

t  H.  A.  Cosvbrsb:  Annals  of  Mathematics,  Ser.  II,  Vol.  V  (1904),  pp.  106-109,  where  is  given  a  fnlle 
explanation  of  these  methods. 


2  Curves  with  a  Directrix. 

Employing  the  notation  used  in  the  memoirs  of  Prof.  Morley,  a  line  a  is 
written  in  the  form 

where  x^  is  the  reflection  of  the  origin  in  the  line,  and  — 1/<„  is  the  clinant. 

The  definition  of  the  first  orthocentre,  as  extended  for  2n — 1  lines,  is  the 
point  of  intersection  of  the  In — 1  perpendiculars,  let  fall  from  the  centres  of  the 
2n  —  1  A^"~"s,  one  of  which  is  uniquely  determined  by  every  2n — 2  lines,  upon 
the  line  left  out  in  each  case.  By  a*  A^"~^  is  meant  a  deltoid  of  order  2n  —  2 
and  class  2n — 3,  having  2n — 3  cusps,  and  containing  the  line  at  infinity  as  a 
special  {In — 4)-fold  line,  being  tangent  to  it  n  —  2  times  at  /  and  n — 2  times  at  /. 
Thus,  in  the  case  of  three  lines,  the  three  A"s  each  of  which  is  determined  by 
two  lines  of  the  three,  are  simply  points,  the  vertices  of  the  triangle  of  the  three 
lines,  and  the  orthocentre  is  the  intersection  of  the  perpendiculars  dropped  from 
the  vertices  upon  the  opposite  sides.  But  in  the  case  of  five  lines,  every  four 
lines  uniquely  determine  a  A',  and  the  perpendiculars  dropped  from  the  centres 
of  these  five  A^'s  upon  the  line  omitted  in  each  case  meet  in  a  point,  which  is 
the  orthocentre  of  the  five  given  lines. 

For  2n  —  1  lines,  the  first  orthocentre*  is  of  the  form 

i'n-i  =  «!  —  ^j  ag  +  A^g  ag  — +  (— )"  ( +  /S'„_2  a„_i  —  S^-i  a,), 

where  Si  stands  for  symmetric  functions  of  the  2n  —  1  <'s,  and  the  a„  are  the 
so-called  characteristic  constants  of  the  lines,  and  in  the  case  of  2« — 1  lines 


2n-l 


x,tl 


2n-l-a 


"« -    2  {t, - t,)  {t, -t,).... (<i -l2„ _7) * 

From  the  results  deduced  in  the   Orthocentric  Properties,  the  equation  of  the 
directrix  for  any  even  number  of  lines,  2n,  may  readily  be  written  : 


—       kjj 


where  Pn-i  is  a  point  of  the  form  of  the  orthocentre  of  2n  —  1  lines,  but  con- 
structed for  a  2n  line,  and  g'„_i  is  its  conjugate.  It  is  evident,  therefore,  that 
the  clinant  of  the  directrix  of  2n  lines  is 


*n  +  l 


*  Orthocentric  Properties  of  the  Plane  n-Line,  p.  5. 


^..  °' 


Curves  with  a  Directrix.  3 

§  1.     Equation  of  the  Curve. 

Suppose  any  even  number  of  lines  are  given,   2n.    They  have  a  fixed 

directrix,  whose  clinant  is  ^^ii.     Allow  one  of  the  2n  lines  to  move  subject  to 

the  condition  that  the  orthocentre  of  every  set  of  2n — 1  lines,  chosen  from  the 
2n,  shall  always  lie  on  this  directrix.  The  problem  is  to  find  the  curve 
enveloped  by  the  moving  line.  Equating  the  clinant  of  the  directrix  to  a  con- 
stant turn  Ilk,  we  have 


whence 

\V  - '!"' 

'Rii't- 

«n 

—  ^fln  +  l  =  0. 

JDUt 

2n 
«n  =    2  ? 

^1  h 


{h-h){t^-h)-.--{h-t^,,y 


and 


Then 


2n 


«»+!-  ^{t,-t,){t,-t,)....{t,-t,,) 


2n 


•Cj  \.*\  fC  1 1       J 


an      l^a,  +  ^-  2u{t,-t,)(t,~t,)....(t,-t,,)-^- 

Considering  as  the  moving  line  the  line  whose  reflection  in  the  origin  is  x^, 
and  whose  clinant  is  — l/<i,  this  last  equation  may  be  written  in  the  form 

for  a?!  and  ^i  are  the  only  variables,     aj  and  a^  are  symbolic,  and  have  meaning 
only  in  combination. 

Now  the  condition  that  the  equation  of  the  line  in  the  form  taken, 

shall  represent  a  line  is  that 

Substituting  from  this  for  t^  in  the  above  equation,  we  obtain 

«r'  y'i  -kx^  yr'  =  {a,y,  +  a,x,y^'-',  (1) 

which  is  the  pedal  of  the  curve  enveloped  by  the  moving  line. 


4  Curves  with  a  Directrix. 

The  equation  of  a  line  in  circular  coordinates  with  clinant  — Ijt  is 

where  Xj  is  the  reflection  of  the  origin  in  the  moving  line,  and  y^  is  its  conjugate. 
The  equation  of  any  line  in  conjugate  coordinates  may  be  written 

where  ^,  iq  and  ^  may  be  considered  as  trilinear  line  coordinates,  having  for 
triangle  of  reference  the  triangle  whose  vertices  consist  of  the  7  and  J  points  on 
the  line  at  inflnity  and  the  origin.     In  order  for  these  two  lines  to  be  identical, 

c  1 

Dividing  equation  (l)  through  by  a;f"~^yf"~^,  and  substituting  —  -  iox  —  and 

—  ^  for  — ,  the  equation  of  the  envelope  of  the  moving  line  is  obtained  in 
the  form 

This  is  a  curve  of  class  in  —  2,  and  since  the  equation  has  no  terms  in  ^  higher 
than  the  first,  contains  the  line  at  infinity  as  a  (2n — 3)-fold  line.  The  2n — 3 
points  of  tangency,  obtained  by  equating 

_^n-l^n-2^;5.p-2jjn-l 

to  zero,  are  on  the  line  at  infinity  in  the  directions  ^  =  0  n  —  2  times,  jy  =  0 
n  —  2  times,  and  j^z^hvi  once,  where  Ijh  is  the  clinant  of  the  directrix.  The 
curve  is  therefore  a  special  and  specially  placed  (2n  —  3)-fold  parabola,  being 
tangent  to  the  line  at  infinity  n  —  2  times  at  /,  n  —  2  times  at  /,  and  once  in  the 
direction  given  by  ^  =  ^  jy. 

Moreover,  since  the  line  at  infinity  enters  as  a  (2n  —  3)-fold  line,  and  the 
equation  is  of  class  2n  —  2,  the  curve  is  a  special  Jonquieres  curve  in  lines. 
Giving  to  this  curve  the  name  ^^"~^,  we  may  state  generally : 

Theorem  l.  Given  2n  lines,  one  of  which  is  allowed  to  move  subject  to  the 
condition  that  the  orthocentre  of  every  2  n  —  1  lines  chosen  from  the  2  n,  shall  always 
lie  on  the  directrix,  the  resulting  curve  is  a  K^^~^. 


Curves  with  a  Directrix.  5 

The  simplest  K^^"^,  obtained  when  n  =  2,  is  evidently 

which  is  the  equation  of  a  parabola,  tangent  to  the  line  at  infinity  in  the  direction 
given  by  ^  =ky;.  Now,  a  parabola,  which  has  for  the  clinant  of  its  directrix  1/k, 
must  have  — 1/k  for  the  clinant  of  the  direction  to  the  point  where  it  is  tangent 
to  the  line  at  infinity,  since  this  point  must  lie  in  a  perpendicular  direction. 
That  is,  in  the  case  of  the  parabola  K^,  the  direction  indicated  by  ^  =  ky!  is  the 
direction  whose  clinant  is  — 1/k.  But  since  all  curves  K^"~^,  including  the 
parabola,  are  tangent  to  the  line  at  infinity  in  the  direction  given  by  ^^ky;, 
this  means  that  they  are  all  tangent  to  the  line  at  infinity  at  the  point,  the  clinant  of 
the  direction  to  which  is  — 1/k. 

If  the  original  2n  lines  are  chosen  so  that  1/k  is  equal  to  — 1,  the 
directrix  is  perpendicular  to  the  axis  of  reals,  and  the  direction  indicated  by 
^  =  ky!,  in  which  all  curves  K^'^-^  are  tangent  to  the  line  at  infinity,  is  the 
direction  of  the  axis  of  reals.  Hereafter,  every  ^^""^,  which  is  tangent  to  the 
line  at  infinity  n — 2  times  at  /,  n  —  2  times  at  J,  and  once  in  the  direction  of  the 
axis  of  reals,  mil  he  called  a  Kl'^~^. 

For  n  =  3  and  ^=  —  1,  we  have  the  general  Jonquieres  curve  Kl,  containing 
the  line  at  infinity  as  its  triple  line.  Thought  of  as  a  triple  parabola,  the  curve 
is  specially  placed,  being  tangent  to  the  line  at  infinity  at  /,  /,  and  in  the  direc- 
tion of  the  axis  of  reals. 

Orthocentre  of  2n  —  1  Lines  of  a  K^^-^, 

Any  one  of  the  2n  lines  might  be  allowed  to  move  subject  to  the  condition 
that  the  orthocentre  of  every  set  of  2n  —  1  lines,  chosen  from  the  2n,  should 
always  lie  on  the  fixed  directrix,  and  the  resulting  curve  would  be  identically 
the  same,  for  the  moving  line  must  at  different  times  coincide  with  each  position 
of  the  2n  —  1  fixed  lines,  since  their  orthocentre  lies  on  the  directrix.  Inasmuch 
as  any  one  of  the  In  lines  might  be  allowed  to  vary  for  a  time,  then  remain 
fixed,  and  one  of  the  others  be  allowed  to  vary,  subject  to  the  given  conditions, 
it  is  evident  that  the  In  —  1  lines  are  any  2n  —  1  lines  of  the  curve.  Hence, 
the  theorem  follows : 

Theorem  2.  The  orthocentre  of  any  2n  —  1  lines  of  a  K^^~^  must  always  lie 
on  the  directrix. 

Thus,  in  the  case  of  the  K^,  where  n  =  3  (that  is,  in  the  case  of  the 
Jonquieres  curve  of  the  fourth  class,  having  the  line  at  infinity  as  a  triple  line), 
the  orthocentre  of  any  five  lines  of  the  curve  must  always  lie  on  its  directrix. 


6  Curves  with  a  Directrix. 

§2.     Some   Theorems   Concerning  the  Orthocentre  of  In  —  1  Lines. 
In  the  Orthocentric  Properties,  already  mentioned,  it  was  proved  that  the 

cr 

first  orthocentre  of  2n  — 1  lines  is  of  the  form     "~^  ,  when  the  lines  are  taken 

as  tangents  of  a  A^"~^  By  ^„_i  is  meant  the  sum  of  the  products  of  the 
In  —  1  <'s,  taken  n  —  1  at  a  time,  where  ijt^  is  the  clinant  of  the  line  a,  and 
similarly  for  S^^n-i-  ^or  the  work  of  this  section  it  is  more  convenient  to  take 
the  equation  of  the  A^"~^  in  the  form 

where  the  clinant  of  any  line  a  is  now  t^ .  Consequently,  the  orthocentre  of 
2n  —  1  lines  tangent  to  a  A^"~^  is  now  Sn. 

Considering  three  lines  tangent  to  a  A^,  which  may  be  any  three  lines  in 
the  plane,  since  a  A^  can  be  taken  tangent  to  any  four  lines,  their  orthocentre 
is  Sz,  where  S  stands  for  symmetric  functions  of  the  three  t^s,  which  are  the 
respective  clinants  of  the  three  lines.     This  may  be  written 

where  S  now  stands  for  symmetric  functions  of  two  Vs.     If  /Si  vanishes;  i.  e.,  if 

k  +  k=  0, 
the  orthocentre  is  evidently  independent  of  the  third  line,  and  is 

—  f  1. 
The  condition  on  the  two  remaining  lines,  since  the  clinant  of  one  is  equal 
to  the  negative  of  the  other,  is  that  they  he  perpendicular.     This  is  nothing  more 
than  the  well-known  case  of  elementary  geometry,  where  the  orthocentre  of  a 
right-angled  triangle  is  at  the  vertex  of  the  right  angle. 

Considering  next  five  lines  tangent  to  a  A^,  which  may  be  any  five  lines  in 
the  plane,  since  a  A^  can  be  taken  tangent  to  any  six  lines,  their  orthocentre 
is  S^,  where  S  stands  for  symmetric  functions  of  the  five  ^'s,  which  are  the 
clinants  of  the  five  lines.     This  may  be  written 

Sz  +  «6  S^  {S  for  4  t's). 

If  S^  vanishes,  the  orthocentre  is  evidently  independent  of  the  fifth  line,  and  is 

/S'a  {S  for  4  ^'s). 

This  may  be  written 

Si  +  hS^  {SiotZt'B). 


Curves  witJi  a  Directrix.  7 

If  /Sj  vanishes ;  that  is,  if 

which  carries  with  it  its  conjugate  relation 

ti  +  h  +  h—  0, 
the  orthocentre  is  evidently  independent  of  the  fourth  line,  and  is 

^  (^^for3<'s); 

that  is,  ti  <2  ^3  • 

Since  ^Si  and  S^  for  three  t's  vanish,  it  follows  that 

and  the  orthocentre  is 

*!• 

That  is,  the  condition  on  the  three  remaining  lines  is  that  they  form  an  equilateral 
triangle. 

Now  the  equation  of  any  line  tangent  to  a  A"  is 

I  —  at  -\-  xt^  —  y t^  -{-  ht*'  —  t'^  =  0. 
The  three  intersections  of  the  three  lines,  which  form  an  equilateral  triangle,  are: 

x,  =  -^+ht\-\.t\, 
n 

h 
x,^-^+h^tl  +  t\. 

Their  centroid  is  tl,  which  is  the  orthocentre.  Therefore,  the  orthocentre  0/  any 
Jive  lines  is  independent  of  two  of  the  lines,  when  the  remaining  three  form  an  equi- 
lateral triangle,  and  is  the  same  as  the  centroid  of  the  three  paints  of  intersection  of 
the  three  lines. 

Now  consider  seven  lines  tangent  to  a  A''^,  which  may  be  any  seven  lines  in 
the  plane,  since  a  A'  can  be  put  tangent  to  any  eight  lines.  Their  orthocentre 
is  Si,  where  iS' stands  for  symmetric  functions  of  seven  t's.     This  may  be  written 

Si  +  t^  S3  {S  for  6  t'a). 

If  S3  vanishes,  the  orthocentre  is  independent  of  the  seventh  line,  and  is 

/Si  {Sror6t'B), 

which  may  be  written 

Si+tA  {S  {or  5  t's). 


8  Curves  with  a  Directrix. 

If  S3  vanishes,  the  orthocentre  is  independent  of  the  sixth  line,  and  is 

/Si  {S  for  5  <'s). 

This  may  be  written 

Si  +  <B  ^3  {S  for  4  t'&). 

If  aS's  vanishes,  the  orthocentre  is  independent  of  the  fifth  line,  and  is 

Si  {S  for  4  Vs)  ; 

that  is,  ti  t^  <3  ti . 

The  vanishing  of  S^ ,  where  S  stands  for  symmetric  functions  of  four  t's,  carries 
with  it  its  conjugate  relation 

Si  =  0  (S  for  4  fa). 

The  vanishing  of  S3  for  five  t's  may  be  expressed : 

S3  +  t,S,=:0  (Sfor^t's). 

But  in  order  for  the  orthocentre  to  be  independent  of  the  fifth  line,  it  was  proved 
that  S3  of  four  t's  must  vanish.     Consequently, 

Sz  =  0  {S  for  4  t's). 

From  these  three  relations, 

Si  =  0,    ^3  =  0,    ^3  =  0         (SfoT^t's), 

it  follows  that  the  four  remaining  lines  form  an  ordered  4-gon  with  equal 

angles  — ,  for  the  relation 

4  •  • 

exists  between  their  clinants.     Consequently,  the  orthocentre  reduces  to 

/4 

The  equation  of  any  line  tangent  to  a  A'  is 

l  —  a2t  +  ait^--xt^  +  yt^  —  bit^  +  b2t^  —  t''  =  0. 
The  four  points  of  intersection  of  the  four  lines  which  form  an  ordered  4-gon 
with  equal  angles-^,  are: 

x,=       ^  +  J^lil^ILlL  +b,itl-b,ii-l)t\-t\, 
tl  ti 

:r,  =  -'!^  +  ^li^=:i^^-b,itj-b,il  +  i)t\-t\, 

0-3=  'b^^   +   ^li-l+l)    +b,itl-b,{-i+l)tl-tt, 

t\  t^ 

a;i    ,      ai{i  - 

7f  +  t, 


x,  =  -^^+     "^(\+^)     -b,iti-b,{-l-i)tl-t\. 


Curves  xjoiih  a  Directrix. 


9 


Their  centroid  is  — 1\,  which  is  the  orthocentre.     Therefore,  the  orihocentre  of 
any  seven  lines  is  independent  of  three  of  the  lines,  when  the  four  remaining  lines 

form  an  ordered  4-gon  with  equal  angles  — ,  and  is  the  centroid  of  the  four  points  at 


n 


which  are  the  angles  — 


7t 


But  in  addition  to  the  four  -  points  of  intersection  of  the  four  lines,  there 


7t 


are  two  points  where  the  lines  intersect  at  the  angle  — .     The  centroid  of  these 

two  points  is  also  the  same  as  the  orthocentre  of  the  seven  lines  where  three  are 
arbitrary,  for  these  points  are 


_  «2 


"ill  C  1  , 


x"=       ^+bitl-t{, 
'■1 

and  their  centroid  is  — 1\,  which  is  the  orthocentre. 


4-gon  with  equal  angles  -j-. 


Passing  to  the  case  of  nine  lines  tangent  to  a  A',  the  orthocentre  jS^,  where 
S  stands  for  symmetric  functions  of  nine  t's,  is  independent  of  the  sixth,  seventh, 
eighth  and  ninth  lines,  when  the  remaining  Jive  lines  form  an  equiangular  pentagon, 
tlie  equal  angles  being  ~,  and  becomes  tl.     As  in  the  other  cases,  this  is  easily 


10  Curves  with  a  Directrix. 

proved  by  calculation  to  be  the  centroid  of  the  five  points  at  which  are  the 
angles  — . 

Now  it  is  evident  that  the  process  we  have  been  carrying  on  can  be  con- 
tinued without  limit,  and  that  the  orthocentre  SnOf  2n  —  1  lines  tangent  to  a  A^"~^ 
is  independent  of  n  —  1  of  the  lines,  and  becomes  ( — 1)""^^",  when  the  remaining 
n  lines  form  an  ordered  polygon  with  equal  angles  — ,  and  that  the  orthocentre  is 

always  the  centroid  of  the  n  -  points  of  intersection  of  these  n  lines. 

But  it  was  observed  in  the  case  of  four  lines  which  formed  an  ordered  4-gon 
with  equal  angles  -,  that  while  the  centroid  of  the  four  -  points  was  the  ortho- 
centre  of  seven  lines,  where  three  were  arbitrary,  the  centroid  of  the  two 
~  points  was  also  the  orthocentre  of  the  seven  lines. 

Passing  to  the  case  of  five  lines  which  form  an  ordered  pentagon  with 
equal  angles  -,  there  are  five  points  where  the  intersecting  lines  make  the 

o 

angle  - ,  and  five  points  where  they  make  the  angle  -—  .     The  centroids  of  both 

these  sets  of  points  are  the  same,  as  is  easily  proved  by  taking  the  five  lines  as 
tangents  of  a  A^,  and  calculating  the  centroids  of  the  two  sets  of  points. 

In  the  case  of  six  lines  which  form  an  ordered  hexagon  with  equal  angles  -r, 

there  are  six  angles  -,  six  angles  -—  and  three  angles  -— ,  and  the  centroid  of 
6  6  6 

each  set  of  points  is  the  same,  as  can  most  easily  be  proved  by  taking  the  six 

lines  as  tangents  of  a  A',  and  calculating  the  centroid  of  each  set  of  points. 

The  general  scheme,  for  convenience  printed  at  the  end  of  this  section,  may 

be  written  down  in  the  case  of  n  lines,  which  form  an  ordered  polygon  having 

equal  angles  — . 

The  fact,  proved  in  the  particular  cases,  concerning  the  centroid  of  the  sets 
of  points  in  the  case  of  n  lines  which  form  an  ordered  polygon  having  equal 
angles  - ,  is  evidently  true  in  general,  and  may  be  verified  in  any  particular 

case,  following  the  given  method  of  proof.     Hence,  the  theorem  follows: 

Theorem  3.     The  centroid  of  any  one  of  the  sets  of  points  associated  with  n  lines 

which  form  an  ordered  polygon  with  equal  angles  -  ,  is  the  centroid  of  all. 

7% 


Curves  with  a  Directrix.  11 

Also,  since  the  centroid  of  the  n  -  points  was  proved  to  be  the  orthocentre 

of  2n — 1  lines,  the  theorem  follows: 

Theorem  4.     The  orthocentre  of  2n — 1  lines  is  independent  of  n  —  1  of  the  lines 

when  the  remaining  n  lines  form  an  ordered  polygon  with  equal  angles  - ,  and  is  the 
centroid  of  any  set  of  points  associated  with  the  n  lines. 

TABLE. 

2  lines  give        1        angle    — = —  . 

3  «  "  3  angles  -^ — .  ^ 

4  «  "  4  "       -^ — ,  2  angles  -^— • 

5  «  "  5  "       —I — ,  5  "  \"     . 

6  «  "  6  «       — ^,  6  «        -^,        3        angles  -^^-  . 

If  tl        n  7  <(  ^  7  ((  '^^  7  «  _r.!! 

•  •  S       J  '  ly       >  '  7         ' 

8  «       «  8  "       -^,        8  «       ^-,        8  «       ^^,    4  angles*^. 

9  «       «  9  «       _|_,        9  '.       -^,        9  «       ^^,    9      "       ^. 

10         "       «        10  "       -^,      10  «        -4J-'      1^  "        -T^'^^      "      r^'^^'^Slesf^. 


2m         "       "        27»  «       ^i^^,      2m  «       -^^,      2m  "        -^^^ 

2m   '  2m  27« 

2  m  angles  ("^~  ■>''    ,«  angles  ^. 
^  2m  "      2m 

(2m  +  l)     «       "    (2m  +  l)     "       3^,  (2-  +  1)     "       ^lll'  (^'"  +  1)     "       2-^1' ' 

(^-  +  1)    "      TOT'  ^^^'^+1^^"^'^^  2^1- 

§  3.      General  Discussion  of  the  K^^. 

In  the  first  section,  the  equation  of  the  JT^""^  was  obtained  in  trilinear 
complex  line  coordinates,  and  the  orthocentre  of  any  2n  —  1  of  its  lines  was 
proved  to  lie  on  the  directrix  of  the  curve.  In  the  present  section  an  equation 
of  the  curve  will  be  derived  by  another  method,  and  a  general  discussion  will 
follow.     Some  special  cases  will  also  be  considered. 


12  Curves  ivith  a  Directrix. 

Envelope  of  a  A^"~^  Moving  Under  Parabolic  Translation. 

Consider  the  general  case  of  a  A^"~^  moving  under  parabolic  translation; 
that  is,  a  translation  in  which  one  and  therefore  all  the  points  of  the  A^"~^ 
describe  a  parabola,  although  not  the  same  parabola.  A  convenient  form  in  which 
to  write  the  map  equation  of  the  general  parabola  for  the  present  purpose  is 


X 


r' 


{r^  —  ty 

where  t  is  a  constant  turn.  This  parabola  is  of  unit  size,  but  since  the  unit  of 
measurement  may  be  taken  arbitrarily,  it  may  be  considered  as  any  parabola. 
On  dividing  the  map  equation  by  its  conjugate,  and  substituting  the  value  of  t 
for  which  x  becomes  infinite,  1/t^  is  found  to  be  the  clinant  of  the  vector  from 
the  origin  to  the  point  at  infinity  on  the  parabola. 

The  map  equation  of  the  general  A^"~^  with  its  centre  at  the  origin  is 

(-)"[na„_ir-^-(n-l)a„_2r-H- •••  +  (-)"(••••  + 2a,  ^  +  ^J-....) 

(n-2)&„_2      (n-l)&^i1 

••••         r=i      ^       r      J' 

The  condition  for  the  envelope  of  the  A^"~^ ,  moving  with  its  centre  on  the 
parabola 


X 


X  ^ 


T^ 


{t^  —  ty 

while  its  orientation  remains  constant,  is  that 

tDtf  ,  .  <  A/ 

— =r-^,  =  a  real  quantity  =  = 

<iA./  ^  t,D^J 

in  the  equation 


X  : 


r^ 


\r'-t 


6i _  \         _  («  — 2)6„_a       («— l)&„_i1 

••')■■••  p-J         "f-  r  J- 


+ 


t^      ' 

This  condition  is  satisfied  when 

t  =  ti. 

Accordingly  the  equation  of  the  envelope  of  a  A^"~^  moving  under  parabolic 

translation,  is 

P^2+  (-r[««n-l<"-'-(«-l)«n-2<"-'+  •  •  •  •  +  (-r(.  •  .  .  +  2a,< 

,   6i_  \  («— 2)&„_a   ■   (^— l)^n-i"| 

T  ^2  )  fn-1  T  ^  J  • 


X 


Curves  with  a  Directrix.  13 

A  vector  from  the  origin  to  any  point  on  the  envelope  can  therefore  be  considered 
as  the  sura  of  two  vectors,  one  from  the  origin  to  the  centre  of  the  A^"~',  and 
the  other  from  the  centre  of  the  A^"~\  which  is  a  point  of  the  parabola,  to  the 
corresponding  point  of  the  A^"~^ 

Since  the  A^"~^  contains  the  line  at  infinity  n  —  1  times  as  an  isolated  double 
line,  and  since  the  parabola  is  tangent  to  it  once  in  the  direction  1/t^,  the 
envelope  must  contain  it  as  a  special  (2n  —  l)-fold  line,  and  may  be  thought  of 
as  a  special  and  specially  placed  {2n  —  l)-fold  parabola,  being  tangent  to  the  line 
at  infinity  n  —  1  times  at  /,  n  —  1  times  at  /,  and  once  in  the  direction  I/tt^^ 
The  envelope  is  of  order  2 «  -|-  2,  as  is  seen  on  substituting  from  the  map 
equation  in  the  equation  of  a  line,  and  it  is  scarcely  necessary  to  remark,  that 
on  account  of  the  singularities  at  /  and  J  in  the  case  of  the  A^"~^  and  conse- 
quently in  the  case  of  the  envelope,  at  each  point  the  n  —  1  points  of  tangency 
are  equivalent  to  but  n  coincident  intersections  of  the  curve  with  the  line 
at  infinity. 

On  multiplying  the  map  equation  of  the  envelope  by  t  and  subtracting  the 
conjugate  equation,  the  relation 

<»-y-^,+ (-r[|^-^^ +••••+(-)"(•  •••+7^-«i^+----) 

....  4- a^2<"-i  — 0^1^1  =  0 

is  obtained.  This  equation  is  self-conjugate,  and  on  eliminating  y  by  partial 
differentiation  with  respect  to  t,  gives  again  the  map  equation.  This  self- 
conjugate  relation  is  therefore  the  line  equation  of  the  envelope.  From  this 
equation,  the  envelope  is  seen  to  be  of  class  2«,  for,  clearing  of  fractions,  the 
parameter  t  enters  to  the  2  nth  power.  Hence,  it  is  perhaps  better  to  think 
of  the  envelope  as  a  special  Jonquieres  curve  of  class  2n.  When  it  has  been 
proved  that  the  orthocentre  of  any  2  n  + 1  of  its  lines  lies  on  a  directrix  whose 
clinant  is  — 1/1'^  it  will  further  be  known  to  be  a  K^'\  To  introduce  the  proof 
of  this,  we  consider  some  special  cases. 

Directrices. 
The  equation  of  the  directrix  of  the  parabola 

_       -r« 


14  Curves  with  a  Directrix. 

as  the  locus  of  two  perpendicular  lines,  is  evidently 

a;  =  - 


t 


2' 


The  line  equation  of  the  envelope  of  a  A'  moving  with  its  centre  on  this 
parabola  while  its  orientation  remains  constant,  is 


T^—t^   t 


t^-y-:,^.  +  -^-a,t^  =  o. 


This  represents  a  general  Jonquieres  curve  of  the  fourth  class,  or,  thought  of 
otherwise,  a  triple  parabola  tangent  to  the  line  at  infinity  at  /,  J,  and  in  the 
direction  1/t^  Now,  from  Theorem  4,  §2,  it  is  known  that  the  orthocentre 
of  any  five  lines  of  the  curve  is  independent  of  two  of  the  lines,  when  the 
remaining  three  form  an  equilateral  triangle,  and  is  the  centroid  of  the  three 

—  points  of  intersection  of  the  three  lines.    The  clinant  of  any  line  is  \/t.     The 

o 

clinants  of  any  three  lines,  which  form  an  equilateral  triangle,  are  l/<,  Ijat, 
1/o^t.     The  three  —points  of  intersection  of  these  three  lines  are  : 

^^=     {r^-t)t'-c.t)    +^V  +  ^^(^+")' 

Their  centroid  is 

i  {«i2  +  x^  +  «3i)  =  :^vz-i3  =  a:,  say. 

This  then  is  the  orthocentre.  Allowing  t^  to  vary,  x  evidently  traces  out  a  line, 
which  is  the  locus  of  the  orthocentres  for  every  five  lines  of  the  curve,  and  which 
is  therefore  the  directrix  of  the  curve.  The  clinant  of  the  directrix  is  the  clinant 
of  the  direction  of  the  point  at  infinity  on  it.  The  point  at  infinity  is  obtained 
from  the  equation  of  the  directrix  when  t^  =  r^.  Hence,  substituting  this  value 
for  t^  in  the  expression 


r" 

X 

= 

~t^ 

-Tt^ 

= 

Curves  with  a  Directrix.  15 


the  clinant  of  the  directrix  is  found  to  be 

1 


Now,  the  line  equation  of  the  envelope  of  the  parabola  and  the  A^  is 

.  t^-y-  J^,-\l  +h-<i.t'  +  a,t'=.0. 


This  represents  a  special  Jonquiekes  curve  of  class  six,  being  tangent  to  the  line 
at  infinity  twice  at  I,  twice  at  J,  and  once  in  the  direction  1  /r^.  The  two  points 
of  tangency  at  I,  and  likewise  at  /,  are  equivalent  to  but  three  coincident  inter- 
sections of  the  curve  with  the  line  at  infinity. 

Now,  from  Theorem  4,  §2,  it  is  known  that  the  orthocentre  ot  any  seven 
lines  of  the  curve  is  independent  of  three  of  the  lines  when  the  remaining  four 
form  an  ordered  4-gon  with  equal  angles  ^,  and  is  the  centroid  of  the  two  -  points 

of  intersection  of  the  four  lines.  The  clinants  of  any  four  lines  of  the  curve 
which  form  an  ordered  4-gon  with  equal  angles  ^  are  Ijt,  \/it,  —lit,  — Ijit. 

One  point  —  is  given  by  the  intersection  of  the  two  lines  whose  clinants  are 

\jt  and  — Ijt.  The  other  —  point  is  given  by  the  intersection  of  the  other  two 
lines,  whose  clinants  are  Ijit  and  — Ijit.     These  two  points  are 

T^  &!  .2 

r^  ,  6i   ,        2 

Their  centroid  is 


X 


t" 


v^  —  t* 


This,  then,  is  the  orthocentre.  Allowing  t*  to  vary,  x  evidently  traces  out  a  line, 
which  is  the  locus  of  the  orthocentres  of  every  seven  lines  of  the  curve.  The 
clinant  of  this  directrix,  found  as  before,  is  — l/r^ 

Similarly,  the  locus  of  the  orthocentre  for  any  nine  lines  of  the  envelope 
of  the  parabola  and  a  A^  is 


X  = 


T» 


and  its  clinant  is  — 1/t^. 


16  Curves  with  a  Directrix. 

From  these  equations,  it  is  evident  that  the  orthocentre  of  any  2n  +  1  lines 
of  the  envelope  of  the  parabola  and  a  A^"~^  lies  on  a  directrix,  whose  equation  is 


X  = 


^8»i  +  l 


2)1  +  2 /n  +  1  ■ 


^2)1  +  2  _^ 

Its  clinant,  calculated  as  above,  is  — 1/t^.  Since  the  equations  of  the  directrices 
all  give  the  same  clinant  — 1/t^,  and  since  the  absolute  value  of  the  reflection  of 
the  origin  in  them,  being  |  r  | ,  is  equal  to  unity,  it  is  evident  that  they  are  all 
the  same.     Hence : 

Theorem  5.  The  envelope  of  a  A^"~^  moving  under  parabolic  translation  is  a 
K^^,  whose  directrix  is  the  same  as  the  directrix  of  the  parabola  described  by  the 
centre  of  the  /^^^~^ . 

Method  of  Plotting  the  K^"". 

The  line  equation  of  the  general  parabola  used  above  is 

tr 
tx—y—  ^_^  =  0. 

The  line  equation  of  the  A^"~^  is 

(_)«  (a;f » _  ytn-l)  -  fn-l  _  1  _  (^,  <2n^Z  _  S^^_,  <)+•••• 

The  line  equation  of  the  K^^  is 


. . .  +  a„_2 1 


n-l 


a„_i  p]  =  0. 


The  tangents  at  corresponding  points  of  these  three  curves  are  parallel,  for  the 
clinant  of  each  is  the  same,  Ijt.  Since  it  was  proved  that  a  vector  from  the 
origin  to  any  point  on  the  envelope  of  the  parabola  and  the  A^"~^  is  equal  to 
the  sum  of  the  vector  from  the  origin  to  the  centre  of  the  A^"~\  and  of  the 
vector  from  the  centre  of  the  A^"~^,  which  is  a  point  on  the  parabola,  to  the 
corresponding  point  on  the  A^"~\  it  is  evident  that  by  drawing  a  tangent  to  the 
^n-i  ^ff\yxc\x  is  parallel  to  the  tangent  to  the  parabola  at  the  centre  of  the  A^*^^, 
a  corresponding  point  of  the  A^"~^  is  thus  obtained,  which  is  a  point  of  the  -£"*". 
As  the  centre  of  the  A^"~'  takes  all  positions  on  the  parabola,  while  its  orien- 
tation remains  constant,  all  points  of  the  ^^"  are  thus  obtained. 


18  Cv/rves  with  a  Directrix. 

Consider  now  the  more  general  question  of  the  envelope  of  a  A^"'~*,  whose 
orientation  is  constant,  and  whose  centre  is  moving  on  a  ^^".     The  condition  for 

the  envelope  is  that 

t  D  f 

.  r!  J-  =  its  conjugate 
h  ^ti  J 
in  the  equation 


X 


+  (— rfma:.,  er'  -  (»  -  l)  a;^,  (,-'  +  •■•• 

+  (-)•>(....+  2a.' >.  +  I  -....)...  -  ('"-ft^--'  +  C"-^) ^•■-]  . 
This  condition  is  satisfied,  as  before,  when 

That  is,  the  equation  of  the  envelope  is  of  the  form 

It  is  evident,  therefore,  that  when  m<^n,  the  equation  reduces  simply  to 

and  is  another  K^,  with  the  same  directrix  as  the  E^^  over  which  it  was  trans- 
lated, for  the  equation  of  the  directrix  is  independent  of  the  constants  of  the 
deltoidal  term. 

When  m^n,  the  equation  of  the  envelope  reduces  to 


"f  j_  A2m-1 

{r^—tf 


n.  —  ^  _1_  A2m— 1 


and  is  evidently  a  K^^  with  the  same  directrix  as  the  iT^"  over  which  it  was 
translated,  since  both  curves  can  be  obtained  by  trao slating  the  proper  deltoid 
over  the  same  parabola,  and,  consequently,  according  to  Theorem  5,  have  the 
same  directrix.     Hence,  we  may  state  the  theorem : 

Theorem  6.  The  envelope  of  a  A""'"'^  whose  orientation  is  constant,  and  whose 
centre  moves  on  a  K^",  is  another  K^"  if  nK^n,  and  is  a  K^'"  if  my-n,  and  the 
directrices  of  all  curves  generated  thus  are  the  same;  namely,  the  directrix  of  the  K^". 


Curves  voith  a  Directrix.  19 

It  should  be  observed  that  any  curve  jK"2"+2  can  be  obtained  by  translating 
a  cycloid 

(n+l)a„P+-^ 

over  a  K^^. 

Now,  in  the  equation  of  the  ^^",  if  l/r^  be  taken  equal  to  unity,  the 
directrix  is  perpendicular  to  the  axis  of  reals,  and  the  curve  is  tangent  to  the 
line  at  infinity  n —  1  times  at  /,  n  —  1  times  at  J,  and  once  in  the  direction  of  the 
axis  of  reals,  and  is  then  designated  by  jff'g",  according  to  the  notation  of  §  1.  The 
equation  then  takes  the  form 

X  =  J~(f  +  (-)"  ["«"-!  *"''  -  (»^  -  1)  «»-2  «""'  +  •  •  •  • 

§  4.     Caustics* 
Any  rational  curve  can  be  represented  by  a  map  equation 

which  of  course  carries  with  it  its  conjugate  equation 

y=m 

and  the  general  form  of  its  caustic  by  reflection,  where  the  incident  rays  are 
parallel  to  the  real  axis,  can  easily  be  derived.  Letting  t"  represent  the  unknown 
clinant  of  the  line  tangent  to  the  curve  at  the  point 

/(O, 

the  equation  of  this  line  can  be  written 

x-f{t)  =  t'\jj-f{t)-], 

which,  for  varying  t,  is  the  line  equation  of  the  curve.  On  dividing  through  by 
t%  and  eliminating  y  by  partial  differentiation  with  respect  to  t,  the  equation 

or 


.^f^e>-'m+'nm 


is  obtained,  which  must  be  the  map  equation  of  the  curve.    But,  by  hypothesis, 

x  =  f{i) 

♦Salmon:  Higher  Plane  Curves.     Heath:   Oeometrical  Optict,     F.  Morley:  On  the  CauHie  of  the  Epicycloid. 


20  Curves  with  a  Directrix. 

is  the  map  equation  of  the  curve.     It  follows,  therefore,  that 

tnt)^i^^'f{t) 

%  z       ♦ 

whence 

~m' 

This  then  is  the  clinant  of  the  tangent  to  the  curve  at  any  point  given  by  t. 
Since  when  the  incident  rays  are  parallel  to  the  real  axis,  the  reflected  rays 
make  twice  the  angle  with  the  axis  of  reals  that  the  tangents  to  the  curve  at  the 
points  of  reflection  make  with  the  axis  of  reals,  it  is  evident  that  the  clinant  of 
any  ray  after  reflection  at  the  point  given  by  t,  is 


K]' 


-ft. 

and  its  equation  is 

which,  as  t  varies,  is  the  line  equation  of  the  caustic  of  the  given  curve.    Since 
the  condition  for  the  cusps  of  the  curve 

obtained  by  equating  or  to  zero,  is 

m = 0, 

which  of  course  carries  with  it  its  conjugate  equation 

/'(O  =  0, 

it  is  evident  further  that: 

Theorem  7.     The  cusps  of  any  rational  curve  lie  on  its  caustic. 

Now,  consider  the  case  of  the  K'^^\  The  condition  for  cusps  is  of  degree 
2n  +  2,  showing  that  the  K^^  has  2n  +  2  cusps,  although  not  necessarily  all  real. 
The  line  equation  of  the  caustic  by  reflection,  formed  as  in  the  general  case,  is 


X 


-    1    A.         [       <'        ,1    (_Y\^^n-i       (n— l)6n_a 

-  t^\y    \{\  —  tf^^  'I  t^-i  -     f.-2     ^  •  •  •  • 

+  (»-l)a„_iP]}), 


Curves  with  a  Directrix.  21 

which  may  be  written  in  the  form 

t'x-y  =  i-Y  ^na„_, <»+»  -  (n  -  l)  (a„_,  +  a„_i)  T 

+  («  -  2)  (a„_3  +  a„_2)  <"-!  -....  +  (— )»  j  .  .  .  .  +  2<»  (aj  +  a,) 

+  ^"^  (V.  +  &„-i)  -  "^11^-  a,  t\ 
Its  map  equation  is 

+  (»-2H"-l)  (a„_3  +  a„_3)  <"-3_. . . .  +  (_)»  I . . . .  +  3<(ai  +  a,) 

(n— 1)(«  — 2),,        ,  ,      \,n(n—l),      ~\ 

showing  that  the  caustic  is  a  cycloidal  curve.     It  is  to  be  observed,  that  in 

forming  the  equation  of  the  caustic  of  the  K^^,  since  the  parabolic  term,  j- t^, 

f 
appeard  on  the  left  side  of  the  equation,  and  its  conjugate,  /..       ...g ,  appeared  on 

the  right  multiplied  by  Ijf,  these  terms  canceled  each  other.  Hence,  since 
the  ^"  is  made  up  only  of  a  parabolic  term  and  a  deltoidal  term  of  class  2n — 1, 
it  is  evident  that  the  caustic  by  reflection  of  the  general  A'^""^  with  centre  at  the 
origin  is  the  same  as  the  caustic  by  reflection  of  the  Kl",  and  is  the  cycloidal  curve  on 
which  lie  the  2n  —  1  cusps  of  the  A"'~^  and  the  2n  +  2  cusps  of  the  A'§". 

In  general,  given  any  self-conjugate  equation,  which  is  a  polynomial  in  t, 
thus: 

G:  ajr— as^-^+agr-"— . . . .  +(—)*( a;^  <""*+! +»<"-* 

-a^+a  <"-'-'+  •  ...)••••  +(-)"-*( b^^.f^'  +  yf-b.t^-^ 

+  ....)....+(-)"(•  •••  +  &3<'-&2<  +  ^)  =  0, 
where  a  and  b  are  conjugates,  it  represents  a  cyclogen,  the  clinant  of  any  one  of 
whose  lines  is  ( — )"l/<"~^*.     Considering  the  incident  rays  as  coming  from  the 
point  at  infinity  on  the  axis  of  reals,  the  clinant  of  the  line  equation  of  the 
caustic  of  any  (7„  is  therefore  1/^201-2*)      q^  writing  out  a  few  caustic  equations 


22  Curves  vyith  a  Directrix. 

for  the  less  complicated  cyclogens,  it  is  readily  observed,  that  the  caustic  equations 
also  always  represent  cyclogens,  and  that  if  y  is  the  coefficient  of  t''  in  the  C„, 
it  is  also  the  coefficient  of  t^  in  the  caustic.  Further,  considering  the  caustic  as 
a  polynomial  in  t,  it  is  observed  that  the  middle  term  is  always  absent,  and  that 
the  class  of  the  caustic  is  equal  to  the  sum  of  the  order  of  the  C„  and  of  the  order 
of  t  in  its  clinant ;  that  is,  the  resulting  caustic  is  of  degree  2n  —  2^  in  t.  Hence, 
the  general  form  of  the  caustic  of  the  general  (7„  may  be  written,  thus: 

aj«i!n-2*_a2«2"-2'=-i  +  a3<2"-2*-^-  ....  +(-)'(n-7^- |)(a;+a,+i)  <'""'*+  •  •  ■  • 

+  0<"-*  +  .  .  .  .  +  {-f{n-h-^{y+^,^,)t'^  +  .  .  .  ■+^,t'-^,t  +  (i,  =  0, 
where  a  and  j8  are  conjugates. 

§5.     Discussion  of  the  K\. 

(a)  Cusps. 
The  map  equation  of  the  K\  is 

The  condition  for  cusps,  obtained  by  equating  ^    to  zero,  was  found  to  be 

(j^.+''-^=0.  (C) 

which  is  a  sextic  in  t,  showing  the  -ff"*  has  six  cusps,  although  not  necessarily 
all  real.  There  are  just  as  many  real  cusps  as  equation  (C)  has  roots  <{  which 
are  real  turns. 

Considering  a  and  h  as  variables,  and  calling  them  x  and  y,  equation  (C) 
may  be  written  - 


{1—tf    '  t' 

For  varying  t,  this  is  the  line  equation  of  the  curve  whose  map  equation  is 

This  curve,  composed  of  two  loops,  is  tangent  to  the  line  at  infinity  in  the 

direction   of  the  real  axis,  has  a  double  point  at 

X  =■  1/4,  cuts  the  axis  of  reals  again  at  the  point 

a;  =  —  1/16,  and  is  a  sextic  in  lines.     Now  a  is  any 

point  in  the  plane,  and  for  any  point  a  there  are  as 

many  f's,  which  are  real  turns,  satisfying  equation 

(C),  as  there  can  be  drawn  real  tangents  to  the  curve 

(A)  from  the  point  a.     It  is  evident  geometrically  that  if 

"^~(i-0*' 


Curves  with  a  Directrix.  23 

where  t  may  take  any  value  from  i  to  — i  (that  is,  if  a  lies  inside  both  loops), 
no  real  tangents  can  be  drawn  to  (A),  and  the  K\  has  no  real  cusps.     But  if 

—     —  1 

a-x-       (i_^)4, 

where  t  may  take  any  value  from  i  to  — i  (that  is,  if  a  lies  on  the  smaller  loop), 
two  real  coincident  tangents  can  be  drawn  to  (A),  and  the  K^  has  two  real  coin- 
cident cusps.     If  ^ 

«>-(]:=7/' 

where  t  may  take  any  value  from  i  to  — i ;  and  if  at  the  same  time 

''<-(T^' 
whei'e  t  may  take  any  value  from  1  to  i  and  from  — i  to  1  (that  is,  if  a  lies  out- 
side the  smaller  loop,  but  inside  the  larger),  two  real  distinct  tangents  can  be 
drawn  to  (A),  and  the  Kl  has  two  distinct  real  cusps.     If 

_     _  1 

where  t  may  take  any  value  from  1  to  i  and  from  — i  to  1  (that  is,  if  a  lies  on 
the  larger  loop),  two  real  distinct  tangents  and  two  real  coincident  tangents  can 
be  drawn  to  (A),  and  the  K\  has  two  distinct  real  cusps  and  two  coincident  real 

'"^P'-     ^f  a  =  a:  =1/4, 

that  is,  if  a  lies  on  the  node  of  (A),  two  distinct  pairs  of  coincident  tangents  can 

be  drawn  to  a,  and  the  Kl  has  two  distinct  pairs  of  real  coincident  cusps.     If 

">-(T:^' 

where  t  may  take  all  values  (that  is,  if  a  lies  outside  both  loops),  four  real 
tangents  can  be  drawn  to  a,  and  the  K\  has  four  distinct  real  cusps.  Consequently, 
it  is  impossible  for  the  K\^  to  have  more  than  four  real  cusps. 

(b)  Conic  on  the  Cusps. 

The  line  equation  of  the  /fj  may  be  written  as  a  polynomial  in  t,  thus: 
at'^  —  t^{x->ra)  +  t'^{x-\-y—\)  —  t{y-\-h)-\-h  =  Q. 
The  invariant  g.;^  of  this  quartic  in  t  is 

^l       (x  +  a)(y-f  fe)  )   {x  +  y-iy ^ 

whose  vanishing  expresses  the  condition  that  four  Vs  be  self-apolar.     Hence, 

the  clinants  of  the  four  tangents  to  the  Kq  from  any  point  on  the  conic 

,       {x  +  a){y  +  h)      {x  +  y--iy_ 
ab  ^  +  ~  0 


The  Kq  with  four  real  cusps. 
24 


Curves  with  a  Directrix.  25 

are  self-apolar.  But  from  a  cusp  of  the  K\  three  tangents  coincide,  and  the  four 
clinants  are  self-apolar  regardless  of  the  position  of  the  fourth  tangent.  There- 
fore, this  conic  passes  through  the  six  cusps  of  the  K\. 

(c)  Nodes. 
From  consideration  of  PLiJCKER's  equations,  we  know  that  the  Kl  has  four 
nodes.    Since  the  equation 

'—s'-r^, +  '-»''=" 

is  of  class  four,  there  are  in  general  four  tangents  to  the  K^  from  any  point  of 
the  plane.  From  a  node,  however,  two  pairs  of  tangents  coincide,  and  so  there 
are  only  two  distinct  tangents.  Hence,  if  the  equation  of  the  Kl  be  written  as 
a  polynomial  in  t,  thus  : 

^'-^'(^+l)  +  l'^''  +  2/-l)-~(2^  +  &)  +  ^  =  0,  (B) 

having  for  roots  <i  (where  i  =  1,  2,  3,  4),  then  at  a  node, 

<i  =  <3 ,  and  t^^  ti. 
Equating  symmetric  functions  of  these  roots  to  the  coefficients  of  (B),  we  have 

20^1=1  +  ^,  (1) 

a?  +  2(r,  =  ^+f^,  (2) 

2«ri(T,  =  2^,  (3) 

.l  =  \,  (4) 

where  0^  =  1^ -\- 1^,  and  az  =  tit^.  Eliminating  (Ti  and  a.^  by  means  of  equations 
1,  3  and  4,  the  equation 


^=\ 


X  +  a 

is  obtained,  which  evidently  represents  two  perpendicular  lines,  passing  through 
the  point  — a,  on  which  the  four  nodes  lie ;  that  is,  two  lines  whose  intersections 
with  the  line  at  infinity  are  apolar  with  /  and  /.  But  there  are  two  other  pairs 
of  lines,  which  are  imaginary,  on  which  the  four  nodes  lie.  Projectively,  the 
points  /,  J  and  K  on  the  line  at  infinity,  where  K  is  the  point  in  the  direction 
of  the  axis  of  reals,  behave  in  a  symmetrical  manner.  Therefore,  it  is  evident 
that  the  intersections  of  one  of  these  pairs  of  lines  with  the  line  at  infinity  are  apolar 
with  J  and  K,  and  that  the  intersections  of  the  other  pair  are  apolar  with  K  and  I. 

JoHsa  Hopkins  Univbrsitt,  January,  1907. 


Vita.. 

Clyde  Shepherd  Atchison  was  born  June  28,  1882,  at  Carnegie,  Penna.  He 
attended  the  public  schools,  and  graduated  from  Carnegie  High  School  in  1899. 
In  September  of  the  same  year  he  entered  Westminster  College,  Penna.,  from 
which  he  graduated  with  the  degree  of  A.  B.  in  June  1903.  For  the  past  four 
years,  he  has  been  a  graduate  student  in  Mathematics,  Physics,  and  Psychology 
in  the  Johns  Hopkins  University. 

January  3,  1907. 


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